Okay, here are some cool hat problems:

1. There are 100 perfectly rational people on an island, each with either a red or a green party hat glued to his head. Each person can see everyone else's party hat, but not his own. No one can communicate with each other. Everyone is informed that everyone is wearing either a red or green hat, and that at least one person is wearing a red hat. Each day at noon, a ship comes around, and the captain will let anyone leave the island if they can correctly guess their hat color; but if they guess incorrectly they die. These guesses are publicly known by everyone after the ship leaves. If there are x people wearing red hats, who will leave the island and when, in terms of x?

2. There are a finite number of people. Before the game starts, they may discuss a plan with each other, but after the game begins they cannot communicate (except through their guesses). After the game begins, they stand in a single file line and have a red or green party hat glued to their head, so that each person can see all and only those hats in front of him. Then the executioner asks each person, starting from the back of the line, what color his hat is, and kills him if he is wrong. Everyone else in the line hears the guess, but not whether it was correct. What plan can they come up with to minimize the number of people that die in the worst case scenario? (Note that not everyone is necessarily acting in the interest of his own survival, just in the interest of minimizing the number of deaths in the worst case scenario.)

3. The situation is the same as in #2, except that there are now a countably infinite number of people, and they cannot hear each other's guesses. Is it possible for them to come up with a plan that guarantees that only a finite number of people die?